Havel Hakimi Theorem. It is equivalent to removing the vertex with the highest degree, and docking that. Our goal is to prove a Havel-Hakimi type algorithm to realize bi-graphical sequences.

The Havel-Hakimi theorem says that we can test if a sequence is graphical by the following procedure: Sort the sequence in decreasing order.
In this video I provide a proof of the Havel-Hakimi Theorem which gives a necessary and sufficient condition for a sequence of non-negative integers to be gr.
Abstract We study the problem of existence of a graph whosse degree sequence is given. It is equivalent to removing the vertex with the highest degree, and docking that. The Havel Hakimi algorithm gives a systematic approach to answer the question of determining whether it is possible to construct a simple graph from a given degree sequence.